Check the commutativity and associativity of the following binary… - Vidyadip

Question

Check the commutativity and associativity of the following binary operations:
'*' on Z defined by a * b = a + b + ab for all a, b ∈ Z.

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Answer

Commutativity: Let $\text{a, b}\in\text{Z}.$ Then,
a * b = a + b + ab
= b + a + ba
= b * a
Therefore,
a * b = b * a, $\forall\ \text{a, b}\in\text{Z}$
Thus, * is commutative on Z.
Associativity: Let $\text{a, b, c}\in\text{Z}.$ Then,
a * (b * c) = a * (b + c + bc)
= a + (b + c + bc) + a(b + c + bc)
= a + b + c + bc + ab + ac + abc
(a * b) * c = (a + b + ab) * c
= a + b + ab + c + (a + b + ab)c
= a + b + ab + c + ac + bc + abc
Therefore,
a * (b * c) = (a * b) * c, $\forall\ \text{a, b, c}\in\text{Z}$
Thus, * is associative on Z.

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